Convergence in Fractional Models and Applications

We consider a fractional Brownian motion with Hurst parameter strictly between 0 and 1. We are interested in the asymptotic behaviour of functionals of the increments of this and related processes and we propose several probabilistic and statistical applications.

We shall consider the statistical problem that consists in observing, instead of X(t), the regularization X ε (t) := 1 ε +∞ −∞ ϕ((t − x)/ε)X(x) dx with ϕ as before and to make inference about σ(•).To achieve this purpose we establish first in section 4.1.1 a convergence result for the number of crossings of X ε (•), using the following theorem (Azaïs and Wschebor (1996)) Theorem 1.1 Let {b α (t), t ∈ R} be the fractional Brownian motion with parameter 0 < α < 1.Then, for every continuous function h π 2 −→ denotes the almost-sure convergence, N bα ε (x) the number of times the regularized process b ε α (•) crosses level x before time 1, Z ε (u) = ε (1−α) ḃε α (u)/σ 2α with σ 2 2α = V ε (1−α) ḃε α (u) , and process bα (x) is the local time in [0, 1] of b α (•) at level x.To show the result quoted above for X ε (u), we shall use the fact that X ε (u) is close to K(b ε α (u)) and Ẋε (u) is close to K(b ε α (u)) ḃε α (u) = σ(K(b ε α (u))) ḃε α (u), and this enables us to prove that π 2 converges almost surely to where X (•) is the local time for X in [0, 1].Now suppose that 1 4 < α < 3 4 , we have the following result about the rates of convergence in Theorem 1.1 proved in section 3.4: there exists a Brownian motion W (•) independent of b α (•) and a constant C α,ϕ such that, Using this last result for 1 2 < α < 3 4 , we can get the same one for the number of crossings of the process X ε (•) and we obtain in section 4.1.1 A similar result can be obtained under contiguous alternatives for σ(•) and provides in section 4.1.2a test of hypothesis for such a function.
We study also the rate of convergence in the following result proved by Azaïs and Wschebor (1996) concerning the increments of the fractional Brownian motion given here as Theorem 1.2.
Theorem 1.2 Let {b α (t), t ∈ R} be the fractional Brownian motion with parameter 0 < α < 1.Then, for all x ∈ R and t ≥ 0 where v 2 2α = V [b α (1)], λ is the Lebesgue measure and N * is a standard Gaussian random variable.
Hence, we can write (1) in an other form say, for all t ≥ 0 We find the convergence rate in (2) for a function f ∈ L 2 (φ(x) dx) where φ(x) dx stands for the standard Gaussian measure.We can formulate the problem in the following way: Suppose that g (N ) (x) = f (x) − E [f (N * )] is a function in L 2 (φ(x) dx), whose first non-zero coefficient in the Hermite expansion is a N , i.e. g (N ) (x) = ∞ n=N a n H n (x), (N ≥ 1) for which g (N ) 2 2,φ = ∞ n=N a 2 n n!.The index N is called the Hermite's rank.Find an exponent a(α, N ) and a process X g (N ) (•) such that the functional defined by S (N ) ε (t) = ε −a(α,N ) t 0 g (N ) (Z ε (u)) du converges in distribution to X g (N ) (t).Note that similar problems have been studied by Breuer and Major (1983), Ho and Sun (1990) and Taqqu (1977) for summations instead of integrals.
The limit depends on the value of α, and as stated in Section 3.1, α = 1 − 1/(2N ) is a breaking point.As pointed out in Section 3.3, if instead of considering the first order increments, we take the second ones, then there is no more breaking points and the convergence is reached for any value of α in (0, 1).
As applications of the previous results, we get in Section 4.2 the following: and in Section 4.3 we get the rate of convergence in Theorem 1.2 and we obtain that for all giving the form of the limit, depending also of x, and suggesting the convergence rate in the case where 1/2 < α < 1.
We observe that all the results quoted above for the fractional Brownian motion, have been considered in Berzin-Joseph and León (1997) for the Wiener process (corresponding to the case where α = 1/2), in Berzin et al. (2001) for the F -Brownian motion, in Berzin et al. (1998) for a class of stationary Gaussian processes and in Perera and Wschebor (1998) for semimartingales.
It is worth noticing that in the case of stationary Gaussian processes the results are quite similar to those obtained in the present article for N = 2.
The paper is organized as follows.In this section we introduced the problems and their applications.In section 2 we state some notations and the hypotheses under which we work.Section 3 is devoted to establish the main results.The applications are developed in section 4. Section 5 contains the proofs.

Hypotheses and notations
Let {b α (t), t ∈ R} be the fractional Brownian motion with parameter 0 < α < 1 (see for instance Mandelbrot and Van Ness (1968)), i.e. b α (•) is a centered Gaussian process with the covariance function For each t ≥ 0 and ε > 0, we define the regularized processes We also define, for a ) We shall use the Hermite polynomials, which can be defined by exp(tx − t 2 /2) = ∞ n=0 H n (x)t n /n!.They form an orthogonal system for the standard Gaussian measure φ(x) dx and, if h Breuer and Major (1983)) gives a simple form to compute the covariance between two L 2 functions of Gaussian random variables.Actually, if ) is a Gaussian random vector such that X and Y are standard Gaussian random variables with correlation ρ then (3) We will also use the following well-known property For 0 < α < 1 − 1/(2N ) or α = 1/2 and N = 1, we shall write where we define ρ Note that for N = 1 and 0 < α < 1/2, since a ) 2 , and for α = 1/2 with N = 1, (σ For N ≥ 1 and 0 ≤ t ≤ 1, define a(α, N ) will be defined later.Throughout the paper, C shall stand for a generic constant, whose value may change during a proof.N * will denote a standard Gaussian random variable.

Convergence for S
The processes b α (•) and W (•) are independent.The convergence taking place in 1) and 2) is in finite-dimensional distributions.
Remark 1: If the coefficients of the function g (N ) verify the condition Chambers and Slud (1989), p.328), the sequence S (N ) ε (•) is tight and the convergence takes place in C[0, 1] for 1) and in C[0, 1] × C[0, 1] for 2).This will be the case for g (N ) a polynomial.

Rate of convergence for S
(1) remark below Theorem 3.3).We can also give the rate of this convergence using the three last theorems.Let consider A = {k : k ≥ 2 and a k = 0}.If A = ∅, we define The exponent d(α, N 0 ) will be defined later.
We have the following corollary.

2 nd -order increments
Instead of considering the first order increments of b α (•), we study the asymptotic behaviour of the second order increments.We also get convergence for the corresponding functionals to a Brownian motion for all the values of α in (0, 1).Thus, if ϕ is now in C 2 instead of in C 1 , we define where we suppose as in Theorem 3.1 that A := {k : k ≥ 2 and a k = 0} is not an empty set and we define N 0 = inf {k : k ∈ A}.With the technics used in Theorem 3.1, we can prove Corollary 3.2.
where we have noted ( σ Remark: where B(•) is again a cylindrical standard Gaussian process with zero correlation independent of b α (•).

Crossings and Local time
Let us define the following random variable where N bα ε (x) is the number of times that the process b ε α (•) crosses level x before time 1 and bα (•) is the local time for the fractional Brownian motion in [0, 1] (see Berman (1970)) that satisfies, for every continuous function h and then by Banach-Kac (Banach (1925) and Kac (1943)), Σ ε (h) can be expressed as Using Theorem 3.1 2)(a) we can get Theorem 3.4.
Remark 2: It can be proved that, under the same hypotheses as in (ii) and for general f , with (2 + δ)-moments with respect to the standard Gaussian measure, δ > 0, even, or odd with Hermite's rank greater than or equal to three, where , we can see the last convergence as a generalization of (ii).
2 but for all 0 < α < 3 4 and we can prove that 1

Estimation of the variance of a pseudo-diffusion.
As is well-known, the process b α (•) is not a semimartingale.Thus we cannot, in general, integrate t 0 a(u) db α (u) for a predictable process a(•).However if the coefficient α is greater than 1 2 , the integral with respect to b α (•) can be defined pathwise as the limit of Riemann sums (see for example the works of Lin (1995) and Lyons (1994)).This allows us to consider, under certain regularity conditions for µ and σ, the "pseudo-diffusion" equations with respect to b α (•) 2 and positive σ.We consider the problem of estimating σ when µ ≡ 0. Suppose we observe instead of dx, with ϕ as in section 2, where we have extended X(•) by means of X(t) = c, if t < 0. It is easy to see that the process X(t) has a local time X (x) in [0, 1] for every level x, in fact we have X (x) = bα (K −1 (x))/σ(x) where K is solution of the ordinary differential equation (ODE), K = σ(K) with K(0) = c.Considering N X ε (x) the number of times that the process X ε (•) crosses level x before time 1 and using Theorem 1.1 we can prove: Moreover, using Theorem 3.4 (ii) we can also obtain the following theorem.
converges stably towards Here, Remark: This type of result was obtained for a class of semimartingales, and in particular for diffusions, in Perera and Wschebor (1998).

Proofs of hypothesis
Now, we observe X ε (•), solution of the stochastic differential equation, for t ≥ 0, , for t < 0 and we consider testing the hypothesis against the sequence of alternatives where dx with ϕ as in section 2. We are interested in observing the following functionals Using Theorem 3.1 2)(a) we can prove Theorem 4.2.
Remark 1: There is a random asymptotic bias, and the larger the bias the easier it is to discriminate between the two hypotheses.
Remark 2: We can consider the very special case h ≡ 1 and σ 0 constant.The limit random variable is Recall that the two terms in the sum are independent.

β-increments
As an application of Theorems 3.1 1), 3.2 (i) and 3.3, we obtain the following corollary. where

Lebesgue measure
Let Thanks to Theorems 3.1 1) and 3.3 we have the following corollary.
where (σ Thanks to Corollary 3.1.1we can give the rate of convergence when 1 2 < α < 1. Indeed for 0 ≤ t ≤ 1 and x ∈ R * , let We have the following corollary.
where σ (2) λ is the same as previous corollary.
5 Proofs of the results

Asymptotic variance of S
(N ) If we let u = εx, we get when x tends to infinity and is bounded from above by C x 2α−2 .Since α < 1 − 1/(2N ) or α = 1/2 and N = 1, ||g (N ) || 2 2,φ < +∞ and |ρ α (x)| ≤ 1, we can use the Lebesgue's dominated convergence theorem to get the result.2 Proof of Theorem 3.1.1) We give the proof for the special case where N = 2 (0 < α < 3/4) to propose a demonstration rather different than in 2)(a).Using the Chaos representation for the increments of the fractional Brownian motion (see Hunt (1951)), we can write making the change of variable x = ελ in the stochastic integral, we get We shall consider the following functional where the function where t := (t 0 , . . ., t k ) and α i , i = 1, . . ., k, are defined by , while c j , j = 1, . . ., k, are real constants.We want to prove that where where M (Z(x)) dx and g (2) First, let us prove the following lemma Lemma 5.1 where (σ Proof of Lemma 5.1.Let n be the integer part of 1 ε , i.e. n := 1/ε .To study the weak convergence of S (2) M (Z(x)) dx.
We consider the following functional where Z (m) (•) is an approximation of Z(•) defined as follows, let ψ defined by and ξ(λ)dλ = 1.We define ξ(m) (λ) = m ξ(mλ) and Then (Z(x), Z (m) (x)) is a mean zero Gaussian vector verifying The covariance for Lemma 5.2 gives the asymptotic value of E S (2) Proof of Lemma 5.2.Let Applying the Schwarz inequality Applying Lemma 4.1 of Berman (1992), which gives the required inequality not exactly for g (2) M but for an Hermite polynomial H l , and Mehler's formula (see Equation (3)), we get Lemma 5.2.
2 Now, we write S (2,m) n,M (t) as {ξ i } i∈N * is a strictly stationary m-dependent sequence (and then strongly mixing sequence) of real-valued random variables with mean zero and strong mixing coefficients (β n ) n≥0 .Furthermore, jn i=1 b 2 i,n = 1 and lim n→+∞ max i∈[1,jn] |b i,n | = 0. On the other hand, as in Rio (1995), defining where Q ξ 1 is the inverse function of t → Pr(|ξ 1 | > t), β(t) = β t the cadlag rate function, β −1 the inverse function of this rate function β.We have This last integral is finite if, and only if, E(ξ 2 1 ) < ∞ (see Doukhan et al. (1994)).
for M ≥ 2, because all the terms are limits of variances hence greater or equal to zero, and for l = 2 we have, by Plancherel's theorem Then, for M ≥ 2 and m ≥ m M and then applying Application 1 (Corollary 1, p.39 of Rio (1995)), we finally get that and by Lemma 5.2, lim m→+∞ sup n E S (2) Dynkin (1988), we proved that S (2) a,M ) 2 ) for M ≥ 2 and then Lemma 5.1 follows.
2 Now since lim applying the Dynkin's result, the proof is completed for the case where N = 2 and 0 < α < 3 4 .Note that this demonstration uses the crucial fact that ρ α belongs to L 2 ([0, ∞[) and so can not be implemented for the other cases.For those cases, Theorem 3.1 1) can be proved using the diagram formula, going in the same way as in Chambers and Slud (1989); indeed for this it is sufficient to adapt the following proof of 2)(a).
2)(a).The following result heavily depends on the N value, known in the literature as the Hermite's rank.Suppose that 1/(2N ) < α < 1 − 1/(2N ).As before, it is enough to prove that , where Furthermore (b α (t 1 ), . . ., b α (t k )) and ( W (t 1 ), . . ., W (t k )) are independent Gaussian vectors.We shall follow closely the arguments of Ho and Sun (1990) with necessary modifications due to the fact that we are considering a non-ergodic situation.Let c 0 , . . ., c k , d 1 , . . ., d k , be real constants, we are interested in the limit distribution of ε,M (t j−1 ) .
To simplify the notation we shall write ε,M (t j−1 ) , then Γ ε (t) is a mean zero Gaussian random variable and where ] is given by Lemma 5.3 whose proof is an easy computation.
Lemma 5.4 Thus we deduce the following Lemma 5.5.
The proof is a direct consequence of Lemma 5.4, according to |t j − s| > 2ε or |t j − s| ≤ 2ε and to s > ε or s ≤ ε.
We want to study the asymptotic behaviour of where ds = ds 1 ds 2 . . .ds r .We use the diagram formula.In this case: where G is an undirected graph with l 1 + l 2 + • • • + l r + m vertices and r + m levels (for definitions, see Breuer and Major (1983), p.431), Γ = Γ(1, 1, . . ., 1, l 1 , . . ., l r ) denotes the set of diagrams having these properties, G(V ) denotes the set of edges of G; the edges w are oriented, beginning in d 1 (w) and finishing in d 2 (w).
To the set Γ belong the diagrams such that the first m levels correspond to the Γ ε (t) variables, ρ is defined as α (s i − s j ), if i and j are in the last r levels, ν ε (s j , t), if the edge w joins the first m levels with the last r levels, 1, otherwise.
We say that an edge belongs to the first group if it links two among the first m levels, and to the second if not.
We shall classify the diagrams in Γ(1, 1, . . ., 1, l 1 , l 2 , . . ., l r ) as in Ho and Sun (1990), p. 1166, calling R the set of the regular graphs and R c the rest.We start by considering R.
In a regular graph, since N > 1, the levels are paired in such a way that it is not possible for a level of the first group to link with one of the second, yielding a factorization into two graphs, both regular, and then m and r are both even.We can show as in Berzin et al. (1998) that the contribution of such graphs tends to .
Using the notations of Ho and Sun (1990), p. 1167, and calling D ε /R c the contribution of the irregular graphs in D ε : Any diagram G ∈ R c can be partitioned into three disjoint subdiagrams V G,1 , V G,2 and V G,3 which are defined as follows.V G,1 is the maximal subdiagram of G which is regular within itself and all its edges satisfy 1 as ε → 0, where q = |V * G,1 (2)|/2.The limit is then O(1).Consider now A ε 2 and define V G,2 to be the maximal subdiagram of G−V G,1 , whose edges satisfy m+1 (2) are the levels of V G,2 .A graph in V G,2 is necessarily irregular, otherwise, it would have been taken into account in A ε 1 .As in Berzin et al. (1998), tends to zero as ε goes to zero.For A ε 3 define We assume now that l 1 , l 2 ,. . .,l r , are fixed by the graph.
where E(V G,3 ) are the edges of V G,3 and ν ε (s, t) = 1 aε(t) k j=0 c j α ε (s, t j ) where α ε (s, t) is given by Lemma 5.4.V * G,3 (2) can be decomposed in two parts, where g(i) is the number of edges in the i-th level not connected by edges to any of the first levels.Furthermore we note where k(i) is the number of edges such that d 1 (w) = i.As in Ho and Sun (1990), p. 1169, we can rearrange the levels in V * G,3 (2) in such a way that the levels of B G are followed by the levels of C G .Within B G and C G , the levels are also rearranged so that those with smaller g(i) come first.We have (2), we have (l i − g(i)) edges coming from levels in the first group and thanks to Lemma 5.5 their contribution to A ε 3 is bounded by C ε β(l i −g(i)) and in total for these levels we get the bound C ε β i∈B G (l i −g(i))+β i∈C G (l i −g(i)) ; now the other terms are of the form: ρ (ε) α (s d 1 (w) − s i ) which are bounded by 1, or of that one: This last integral can be bounded by hence, by ( 5) and since l i ≥ N , then we have We have the following bounds The last inequality is obtained by the same argument for showing (27) in Ho and Sun (1990), p. 1170.6) tends to zero with ε.

Second case: |B
(2)|/2 A ε 2 tends to zero with ε and this gives the required limit.

Third case: |B
(1)| > 0 (otherwise it would have been taken in account before in V G,2 ) and ( 6) tends to zero with ε. u), and and the result follows by 2)(a).
Remark 3 also follows from the fact that a 1 Proof of Proposition 5.2.We suppose t > 0. As in Proposition 5.1, we use Mehler's formula and we make the change of variable u = εx to get , when x tends to infinity, and since ||g (N ) || 2 2,φ < +∞, we have with x large enough, the result follows. 2 Proof of Theorem 3.2.From Proposition 5.2 we prove that = 0 and the result is an adaptation of Theorem 3.1 2)(a). 2 Proof of Proposition 5.3.We suppose t > 0 and then t ≥ 4ε.As in Propositions 5.1 and 5.2, we use Mehler's formula and we break the integration domain into two intervals: [0, 4ε] and [4ε, t].
For the first one, making the change of variable u = εv, we get Now, let us have a closer look to the second interval Using a second order Taylor's expansion of (u − εx) 2α in the neighborhood of x = 0, it becomes Then, since 1 − 1/(2N ) < α and ||g (N ) || 2 2,φ < ∞, we can apply the Lebesgue's dominated convergence theorem and the limit is given by the first term in the sum. 2 Proof of Theorem 3.3.Define and from the previous calculations and Lebesgue's theorem D (N ) (ε) → D (N ) (0) as ε → 0 in the L 2 -norm with respect to Lebesgue's measure and Theorem 3.3 follows. 2 A straightforward calculation shows that the second order moment of the first term above is O(ε 2(α−d(α,N 0 )) ) = O(ε).So 1. (i), (ii) and (iii) follows by Theorem 3.1 1), 3.2 (i) and 3.3.
and this concludes the proof of the corollary.2

Asymptotic behaviour of the second order increments
Proof of Corollary 3.2.The proof of corollary follows by using the technics developed in the proof of Theorem 3.1 1) and 2)(a).We just give a sketch of this proof.We can show that , when x tends to infinity, so it holds that ρ α ∈ L 1 ([0, ∞[) and furthermore +∞ 0 ρ α (x) dx = 0, for all α ∈ (0, 1), so (i) follows.In case of the first order increments we required ), does not contribute to the limit because tending to zero in L 2 , so (ii) follows.For the first order increments the bound was ε β with β = inf{α, 1 − α} (see Lemma 5.5 in the proof of Theorem 3.1) and we required β > 1/(2N ) and N > 1 to obtain independence between the limit processes.2 5.2 Some particular functionals

Crossings and Local time
We have to prove the result corresponding to crossings.Recall that Proof of Theorem 3.4.The proof follows along the lines of Berzin et al. (1998) and uses Theorem 3.1 2)(a).By Banach-Kac (Banach (1925) and Kac (1943)), Σ ε (h) can be expressed as where ).We will show on the one hand, that under hypotheses of (ii) and if 0 < α < 3 4 (not only for 1 4 < α < 3 4 as is required for the theorem), and on the other hand, if α ∈ (0, 1) (instead of 0 Moreover we can show that equality (7) is true under the less restrictive hypotheses: h ∈ C 2 with | ḧ(x)| ≤ P (|x|) when 0 < α ≤ 1 2 and furthermore in case where α ≥ 1 2 , we will show that E [S 2 2 ] = o(ε).Thus the term S 2 only matters when α < 1 4 and we will prove in this case that, lim ε→0 ε Let us look more closely at S 1 .
We decompose S 1 into two terms with M big enough.Using Hölder's inequality it's easy to see that Let us see
Applying the change of variable v = u + εx, one has where where Y (u) and Z x (u) are standard Gaussian variables with correlation ρ α (x); furthermore the Gaussian vector Y (u), Z x (u) is independent of b α (u).
Using the Lebesgue's dominated convergence theorem one obtains Let us look at
We now fix (u, v) ∈ C ε and we consider the change of variables with (Z 1,ε (u, v), Z 2,ε (u, v)) a mean zero Gaussian vector independent of (Z ε (u), Z ε (v)) and where α ε (u, v) is given by Lemma 5.4, ρ and Writing the Taylor development of h one has, ).We can decompose Cε as the sum of twenty five terms.We use the notations J j 1 ,j 2 for the corresponding integrals, where j 1 , j 2 = 0, . . .,4 are the subscripts involving h (j 1 ) and h (j 2 ) .We only consider J j 1 ,j 2 with j 1 ≤ j 2 .Then we obtain the followings (A) One term of the form Making the change of variable u − v = εx and applying the Lebesgue's dominated convergence theorem we get (B) Two terms of the form J 0,1 ≡ 0 by a symmetry argument: if L(U, V ) = N (0, Σ) then E U g (2) (U )g (2) (V ) = 0.
(E) Two terms of the form Using (10), one obtains Using the same type of arguments as for (C), (D), (E) we can prove that the other terms are all o(ε).
Using the Lebesgue's dominated convergence theorem we get Now let us consider S 2,1 .
Applying the Lebesgue's dominated convergence theorem we get, with the same notations as before and then Using ( 13) and (15), we have then proved where C ε was defined before.It is obvious that Now we look at K 1 .We fix u and v and consider the change of variables with (Z 3 , Z 4 ) standard Gaussian vector independent of (B ε (u), B ε (v)).
A simple calculus gives Furthermore we have the following limits where p Bε(u),Bε(v) (x, y) stands for the density of vector (B ε (u), B ε (v)) in (x, y).
Writing the third order Taylor development for h one has ḣ with θ 1 between α 3 z and (α 1 x + α 2 y + α 3 z) and θ 2 between (β 3 z + β 4 w) and (β Therefore K 1 is decomposed as the sum of nine terms. (A) One term of the type By ( 17) and ( 18), we get (B) Two terms of the type ×p Bε(u),Bε(v) (x, y) φ(z) φ(w) dz dw dx dy du dv, and this term is zero by Mehler's formula (3).
and then using ( 13), ( 21) and ( 22) we proved that Now let us achieve the proof of (i), proving that ε −2α S 2 For this we write the second order Taylor development for h and we study the two inner corresponding integrals, and doing the same computations as before it yields (i).Now, to achieve the proof of the theorem, we consider, for 1 4 < α < 3 4 , a discrete version of We know by Theorem 3.1 2)(a) that Z n ε (h) → Z n (h), weakly as ε → 0. On the other hand Z 2 n (h) is a Cauchy sequence in L 2 (Ω), this implies that there exists a r.v.Y (h) ∈ L 2 (Ω) such that Z 2 n (h) → Y (h) in L 2 (Ω) as n → ∞; furthermore, we can characterize this variable using the asymptotic independence between b α (•) and Ŵ (•), say To finish the proof it is enough to show Such a proof goes on using the same technics that we have implemented above, for the asymptotic of the second moment.2

Pseudo-diffusion
Estimation of the variance of a pseudo-diffusion.
Proof of Proposition 4.1.We just give an outline of the proof showing that it is enough to consider the fractional Brownian motion case.Because b α (t) has zero quadratic variation when α > 1 2 , it turns out that when σ ∈ C 1 and µ ≡ 0 the solution for the stochastic differential equation can be expressed as X(t) = K(b α (t)), for t ≥ 0, where K(t) is the solution of the ordinary differential equation K(t) = σ(K(t)); K(0) = c.
(for t < 0, X(t) = c).Using the Banach-Kac formula (Banach (1925) and Kac (1943)) we have π 2 where θ is a point between b α (t − εv) and b ε α (t).A similar proof can be done for 0 ≤ t < ε.Hence, we get ε − In a similar way, for 0 ≤ t < ε, we can prove that this expression is bounded by C ε α−δ .Thus, multiplying the last expression by ε −(α+ 1 2 ) it holds uniformly in t ε ( 1 2 −α) ( Ẋε (t) − K(b ε α (t)) ḃε α (t)) = o(1) where Ẋε (u) du and Now, let us study L 1 and L 2 .For L 1 , we have where θ is a point between X ε (u) and K(b ε α (u)) and then Therefore, we can conclude that the asymptotic behaviour of is equivalent to the asymptotic behaviour of Σ ε (h As an application of our result for the fractional Brownian motion in Theorem 3.4 (ii), this term converges stably towards This equation completes the proof. 2 Remark: We conjecture that the same type of result holds for µ = 0, but for the moment we do not have a proof of this statement.

Conclusion
It is interesting to pinpoint the main idea of our methods based on the Gaussian structure of the underlying processes and remark the similarity of the limits obtained in different models considered in Berzin-Joseph and León (1997), Berzin et al. (1998) or Berzin et al. (2001).Also note that our technics allow us the parameters estimation and the setup of tests of hypothesis when the partition is finer.
Following the same approach, future investigations will be made in a more general setup where a drift is introduced in the model.Other results are expected, related with the second order increments, to estimate the Hurst parameter α using variation technics.
dx and B(•) is a cylindrical standard Gaussian process with zero correlation independent of b α (•).The symbol D −→ means weak convergence in finite-dimensional distributions.